Consider a long rod of radius .alpha. impacted by a plane wave of intensity I.sub.0 and frequency .omega. traveling perpendicular to the rod axis, as depicted in FIG. 1. The spatial part of the wave pressure, p.sub.i, can be expanded in cylindrical harmonics as: ##EQU1## .rho. is the fluid density and c is the sonic velocity in the fluid. The radial velocity u.sub.i corresponding to the plane wave expansion is: ##EQU2##
The effect of a rod with its axis located at r=0 is to distort the incoming plane wave, thereby inducing a scattered outgoing wave of a size and shape sufficient to meet the zero radial velocity boundary condition at the rod surface: EQU u.sub.i (k.alpha.)=-u.sub.s (k.alpha.) (4)
where the scattered wave radial velocity u.sub.s is: ##EQU3## and the scattered wave pressure p.sub.s is: ##EQU4## The Hankel function of the first kind: EQU H.sub.m (kr)=J.sub.m (kr)+iY.sub.m (kr) (7)
is descriptive of outgoing waves, valid for r.gtoreq..alpha.. The boundary condition (4) requires: ##EQU5## which must be satisfied term-by-term. Therefore: ##EQU6## The ratio of the derivatives results in a complex function of k.alpha., which can be written as: ##EQU7## These phase shifts are the key to the behavior of the scattered wave and, therefore, to the total intensity pattern of the ultrasonic field.
The total field intensity is given by: EQU I(r,.PSI.)&lt;1/2Re{p.sub.t u.sub.t *}=1/2Re{[p.sub.i +p.sub.s ][u.sub.i +u.sub.s ]*} (12)
where the factor of 1/2 is due to a time-average over a period of the wave. Evidently, computation of the phase shifts allows calculation of the total intensity as a function of spatial position relative to the axis of the rod. The direction .PSI.=0 is defined to be that of the incoming wave propagation vector, as shown in FIG. 1.
The normalized total intensity for a frequency of 213.2 kHz, a rod radius of 6.13 mm and a field radius of 23 mm (k.alpha.=5.3, kr=31.4) is shown in a polar plot in FIG. 2. The incident wave propagates from left to right along the axis of symmetry. The rod axis is perpendicular to the plane of the page and is centered at the origin.
The back-scatter is roughly isotropic, the side-scatter is small, and the forward-scatter has symmetrical lobes, the largest of which are centered at about .+-.25.degree. from the X axis. A significant amount of energy is scattered along the (forward) X axis. The oblique forward and back lobes are the result of edge diffraction from the circumference of the rod. The on-axis back lobe is the specular reflection signal, and the on-axis forward lobe results from incomplete interference effects from the edge diffraction.
A significant amount of energy diffracts around the rod and is redistributed in polar angle, accounting for the forward lobes. This can be understood by examining the scattered wave intensity shown in FIG. 3. Note the broad forward lobe that interferes with the incoming plane wave to form the structured pattern of the total intensity. The back-scatter is rather isotropic, but lower in magnitude than the forward scatter.
Other cases of potential interest exist for different frequencies and rod parameters. The lower frequencies tend to broaden the forward lobes, increase the amplitudes of the diffracted waves and slightly reduce the forward peaking. The higher frequencies have the opposite effects. The complexity of the total field increases with frequency. Structure in the back-scatter increases as more energy is diffracted forward to further interfere with the incident wave. The oblique forward and back lobes are similar to diffraction rings from the cylinder outer edges. The scattered intensity is more peaked forward, and the lobes are more numerous for the higher frequencies.
The case of two or more rods is much more complex than that of an isolated rod, not only because of the boundary conditions, but also because of the issue of coherence. If two widely separated rods are excited by a plane wave impacting each at different times, they scatter the incident energy independently, in general. The resulting field is a superposition of the individual scattered waves, and the intensity is roughly proportional to the algebraic sum of the individual intensities at any particular point. This is also the case when the scatterers are small and fairly closely spaced, but occupy a region of space large compared to the wavelength.
In certain directions, the scattered waves add in phase; in other directions, destructive interference occurs, either partially or totally. This depends on the center-to-center spacing of the rods, as well as their radius and the distance of each from the field point. Furthermore, the time at which each is impacted by the plane wave should be very nearly the same to establish temporal coherence.
In the long wavelength limit, the rods are excited simultaneously. This is not true, in general, for wavelengths comparable with the rod spacing, since the waves travel at the velocity of sound c. The time-of-flight between scattering centers becomes significant in this case, generally destroying coherence. The result is incoherent, or diffuse, scattering in the case of N individual scatterers, and the intensity is proportional to N. Incoherent scattering is quite isotropic for a large number of randomly sized and spaced scatterers.
The scatterers may be arranged in a regular pattern, or array, that is large compared to the wavelength, but having a spacing and radius that is small compared to the wavelength. In this case, coherent scattering may occur in which the intensity in certain directions is proportional to N.sup.2. This is due to a cooperative portion of the scattered waves, in which each individual scatterer adds its contribution to the total wave amplitude in phase. The rest of the scattered intensity is incoherent and appears as a diffuse background radiation. However, because of the long wavelength, no information is available regarding any individual scatterer. In fact, the familiar case of coherent x-ray scattering by crystals yields information about the crystal structure (planar spacing, etc.), but very little else.